3.2621 \(\int \frac{1}{x \left (a+b x^n\right )^2} \, dx\)

Optimal. Leaf size=39 \[ -\frac{\log \left (a+b x^n\right )}{a^2 n}+\frac{\log (x)}{a^2}+\frac{1}{a n \left (a+b x^n\right )} \]

[Out]

1/(a*n*(a + b*x^n)) + Log[x]/a^2 - Log[a + b*x^n]/(a^2*n)

_______________________________________________________________________________________

Rubi [A]  time = 0.064068, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{\log \left (a+b x^n\right )}{a^2 n}+\frac{\log (x)}{a^2}+\frac{1}{a n \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x^n)^2),x]

[Out]

1/(a*n*(a + b*x^n)) + Log[x]/a^2 - Log[a + b*x^n]/(a^2*n)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 9.59624, size = 34, normalized size = 0.87 \[ \frac{1}{a n \left (a + b x^{n}\right )} + \frac{\log{\left (x^{n} \right )}}{a^{2} n} - \frac{\log{\left (a + b x^{n} \right )}}{a^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(a+b*x**n)**2,x)

[Out]

1/(a*n*(a + b*x**n)) + log(x**n)/(a**2*n) - log(a + b*x**n)/(a**2*n)

_______________________________________________________________________________________

Mathematica [A]  time = 0.070419, size = 34, normalized size = 0.87 \[ \frac{\frac{a}{a n+b n x^n}-\frac{\log \left (a+b x^n\right )}{n}+\log (x)}{a^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*(a + b*x^n)^2),x]

[Out]

(a/(a*n + b*n*x^n) + Log[x] - Log[a + b*x^n]/n)/a^2

_______________________________________________________________________________________

Maple [A]  time = 0., size = 45, normalized size = 1.2 \[{\frac{\ln \left ({x}^{n} \right ) }{{a}^{2}n}}-{\frac{\ln \left ( a+b{x}^{n} \right ) }{{a}^{2}n}}+{\frac{1}{an \left ( a+b{x}^{n} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a+b*x^n)^2,x)

[Out]

1/n/a^2*ln(x^n)-ln(a+b*x^n)/a^2/n+1/a/n/(a+b*x^n)

_______________________________________________________________________________________

Maxima [A]  time = 1.43452, size = 58, normalized size = 1.49 \[ \frac{1}{a b n x^{n} + a^{2} n} - \frac{\log \left (b x^{n} + a\right )}{a^{2} n} + \frac{\log \left (x^{n}\right )}{a^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^n + a)^2*x),x, algorithm="maxima")

[Out]

1/(a*b*n*x^n + a^2*n) - log(b*x^n + a)/(a^2*n) + log(x^n)/(a^2*n)

_______________________________________________________________________________________

Fricas [A]  time = 0.226626, size = 68, normalized size = 1.74 \[ \frac{b n x^{n} \log \left (x\right ) + a n \log \left (x\right ) -{\left (b x^{n} + a\right )} \log \left (b x^{n} + a\right ) + a}{a^{2} b n x^{n} + a^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^n + a)^2*x),x, algorithm="fricas")

[Out]

(b*n*x^n*log(x) + a*n*log(x) - (b*x^n + a)*log(b*x^n + a) + a)/(a^2*b*n*x^n + a^
3*n)

_______________________________________________________________________________________

Sympy [A]  time = 3.65584, size = 160, normalized size = 4.1 \[ \begin{cases} \tilde{\infty } \log{\left (x \right )} & \text{for}\: a = 0 \wedge b = 0 \wedge n = 0 \\\frac{\log{\left (x \right )}}{a^{2}} & \text{for}\: b = 0 \\- \frac{x^{- 2 n}}{2 b^{2} n} & \text{for}\: a = 0 \\\tilde{\infty } \log{\left (x \right )} & \text{for}\: b = - a x^{- n} \\\frac{\log{\left (x \right )}}{\left (a + b\right )^{2}} & \text{for}\: n = 0 \\\frac{a n \log{\left (x \right )}}{a^{3} n + a^{2} b n x^{n}} - \frac{a \log{\left (\frac{a}{b} + x^{n} \right )}}{a^{3} n + a^{2} b n x^{n}} + \frac{a}{a^{3} n + a^{2} b n x^{n}} + \frac{b n x^{n} \log{\left (x \right )}}{a^{3} n + a^{2} b n x^{n}} - \frac{b x^{n} \log{\left (\frac{a}{b} + x^{n} \right )}}{a^{3} n + a^{2} b n x^{n}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(a+b*x**n)**2,x)

[Out]

Piecewise((zoo*log(x), Eq(a, 0) & Eq(b, 0) & Eq(n, 0)), (log(x)/a**2, Eq(b, 0)),
(-x**(-2*n)/(2*b**2*n), Eq(a, 0)), (zoo*log(x), Eq(b, -a*x**(-n))), (log(x)/(a
+ b)**2, Eq(n, 0)), (a*n*log(x)/(a**3*n + a**2*b*n*x**n) - a*log(a/b + x**n)/(a*
*3*n + a**2*b*n*x**n) + a/(a**3*n + a**2*b*n*x**n) + b*n*x**n*log(x)/(a**3*n + a
**2*b*n*x**n) - b*x**n*log(a/b + x**n)/(a**3*n + a**2*b*n*x**n), True))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + a\right )}^{2} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^n + a)^2*x),x, algorithm="giac")

[Out]

integrate(1/((b*x^n + a)^2*x), x)